KlappCK

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math-NThis user can contribute with fluent mathematical skills.

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Theorem statement

Suppose $y$ is defined as a function of $x$ by an equation of the form

$f(x)=y\,$

where $f$ is analytic at a point $c$ and $f'(c)$ ≠ 0. Then it is possible to invert or solve the equation for $x$:

$x=f^{-1}(y)\,$

on a neighbourhood of $f(c)$, where $f^{-1}(c)$ is analytic at the point $f(c)$. This is also called reversion of series.

The series expansion of $f^{-1}$ is:

$f^{-1}(y)=c+\sum _{n=1}^{\infty }\left(\lim _{x\to c}\left({\frac {(y-f(c))^{n}}{n!}}{\frac {\mathrm {d} ^{\,n-1}}{\mathrm {d} x^{\,n-1}}}\left({\frac {x-c}{f(x)-f(c)}}\right)^{n}\right)\right).$